The maximal discrete extension of the Hermitian modular group

Abstract

Let ${\Gamma }_n({\mathcal{O}}_{\mathbb{K}})$ denote the Hermitian modular group of degree $n$ over an imaginary-quadratic number field $\mathbb{K}$. In this paper we determine its maximal discrete extension in $\mathrm{SU}(n,n;\mathbb{C})$, which coincides with the normalizer of ${\Gamma }_n({\mathcal{O}}_{\mathbb{K}})$. The description involves the $n$-torsion subgroup of the ideal class group of $\mathbb{K}$. This group is defined over a particular number field ${\hat{\mathbb{K}}}_n$ and we can describe the ramified primes in it. In the case $n=2$ we give an explicit description, which involves generalized Atkin-Lehner involutions. Moreover we find a natural characterization of this group in $\mathrm{SO}(2,4)$.